Furthermore, in order to employ an interventionist account of causation we will need to be able to assign probabilities to measurement outcomes conditional on the actions taken by various possible agents; and although it is straightforward to write down such probabilities, it is less clear what the probabilities actually mean, because if we to keep the agents in coherent superposition then we will not be able to invoke a collapse interpretation to make sense of them. Perhaps the most well-developed extant approach to making sense of quantum probabilities in a no-collapse setting is the Deutsch-Wallace decision-theoretic analysis[1]}, [2]} developed in the context of the Everett interpretation; but this approach may not be applicable in the context of experiments probing indefinite causal order, because it's designed to be applied in the kind of regime where different branches of the wavefunction have effectively decohered and thus interference between branches is impossible, and this is not the case in experiments like the quantum SWITCH because the point of these experiments is that the branches are kept in coherent superposition and ultimately recombined (otherwise we could not verify that the superposition of causal structure has taken place). For example, one important axiom in the decision-theoretic analysis is `Branching Indifference: An agent doesn’t care about branching per se: if a certain measurement leaves his future selves in N different macrostates but doesn’t change any of their rewards, he is indifferent as to whether or not the measurement is performed'[2]} - this axiom seems very reasonable if branches can't interfere, but not so much in the case where agents remain in coherent superposition, because in that case branching brings with it the possibility of phenomena like interference and recombination which could lead to the agents' memories being altered or wiped, and these certainly seem like things they might care about! So it is unclear that we can define a sensible notion of probability for agents involved in experiments like the quantum SWITCH.
